Theorem imaiun1 38469

Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)

Assertion

Ref	Expression
imaiun1	⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3377	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2		vex 3354	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	elima3 5613	. . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
4	3	rexbii 3189	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5		eliun 4659	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
6	5	anbi2i 609	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
7		r19.42v 3240	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
8	6, 7	bitr4i 267	. . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1924	. . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
10	1, 4, 9	3bitr4ri 293	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
11	2	elima3 5613	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
12		eliun 4659	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
13	10, 11, 12	3bitr4i 292	. 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶))
14	13	eqriv 2768	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

Syntax hints:   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∃wrex 3062  ⟨cop 4323  ∪ ciun 4655   “ cima 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-iun 4657  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263

This theorem is referenced by:  trclimalb2  38544